Intrinsic heterogeneity in oscillatory dynamics limits correlation-induced neural synchronization

Optimal phase-selective entrainment of electrochemical oscillators with different phase response curves

We investigate the entrainment of electrochemical oscillators with different phase response curves (PRCs) using a global signal: the goal is to achieve the desired phase configuration using a minimum-power waveform. Establishing the desired phase relationships in a highly nonlinear networked system exhibiting significant heterogeneities, such as different conditions or parameters for the oscillators, presents a considerable challenge because different units respond differently to the common global entraining signal. In this work, we apply an optimal phase-selective entrainment technique in both a kinetic model and experiments involving electrochemical oscillators in achieving phase synchronized states. We estimate the PRCs of the oscillators at different circuit potentials and external resistance, and entrain pairs and small sets of four oscillators in various phase configurations. We show that for small PRC variations, phase assignment can be achieved using an averaged PRC in the control design. However, when the PRCs are sufficiently different, individual PRCs are needed to entrain the system with the expected phase relationships. The results show that oscillator assemblies with heterogeneous PRCs can be effectively entrained to desired phase configurations in practical settings. These findings open new avenues to applications in biological and engineered oscillator systems where synchronization patterns are essential for system performance.

Fast-spiking interneuron detonation drives high-fidelity inhibition in the olfactory bulb

Inhibitory circuits in the mammalian olfactory bulb (OB) dynamically reformat olfactory information as it propagates from peripheral receptors to downstream cortex. To gain mechanistic insight into how specific OB interneuron types support this sensory processing, we examine unitary synaptic interactions between excitatory mitral and tufted cells (MTCs), the OB projection cells, and a conserved population of anaxonic external plexiform layer interneurons (EPL-INs) using pair and quartet whole-cell recordings in acute mouse brain slices. Physiological, morphological, neurochemical, and synaptic analyses divide EPL-INs into distinct subtypes and reveal that parvalbumin-expressing fast-spiking EPL-INs (FSIs) perisomatically innervate MTCs with release-competent dendrites and synaptically detonate to mediate fast, short-latency recurrent and lateral inhibition. Sparse MTC synchronization supralinearly increases this high-fidelity inhibition, while sensory afferent activation combined with single-cell silencing reveals that individual FSIs account for a substantial fraction of total network-driven MTC lateral inhibition. OB output is thus powerfully shaped by detonation-driven high-fidelity perisomatic inhibition.

Theta oscillons in behaving rats

Recently discovered constituents of the brain waves-the oscillons -provide high-resolution representation of the extracellular field dynamics. Here we study the most robust, highest-amplitude oscillons that manifest in actively behaving rats and generally correspond to the traditional θ -waves. We show that the resemblances between θ -oscillons and the conventional θ -waves apply to the ballpark characteristics-mean frequencies, amplitudes, and bandwidths. In addition, both hippocampal and cortical oscillons exhibit a number of intricate, behavior-attuned, transient properties that suggest a new vantage point for understanding the θ -rhythms' structure, origins and functions. We demonstrate that oscillons are frequency-modulated waves, with speed-controlled parameters, embedded into a noise background. We also use a basic model of neuronal synchronization to contextualize and to interpret the observed phenomena. In particular, we argue that the synchronicity level in physiological networks is fairly weak and modulated by the animal's locomotion.

Theta oscillons in behaving rats

Recently discovered constituents of the brain waves-the oscillons-provide high-resolution representation of the extracellular field dynamics. Here we study the most robust, highest-amplitude oscillons that manifest in actively behaving rats and generally correspond to the traditional θ -waves. We show that the resemblances between θ -oscillons and the conventional θ -waves apply to the ballpark characteristics-mean frequencies, amplitudes, and bandwidths. In addition, both hippocampal and cortical oscillons exhibit a number of intricate, behavior-attuned, transient properties that suggest a new vantage point for understanding the θ -rhythms' structure, origins and functions. We demonstrate that oscillons are frequency-modulated waves, with speed-controlled parameters, embedded into a noise background. We also use a basic model of neuronal synchronization to contextualize and to interpret the observed phenomena. In particular, we argue that the synchronicity level in physiological networks is fairly weak and modulated by the animal's locomotion.

Altered patterning of neural activity in a tauopathy mouse model

Alzheimer's disease (AD) is a complex neurodegenerative condition that manifests at multiple levels and involves a spectrum of abnormalities ranging from the cellular to cognitive. Here, we investigate the impact of AD-related tau-pathology on hippocampal circuits in mice engaged in spatial navigation, and study changes of neuronal firing and dynamics of extracellular fields. While most studies are based on analyzing instantaneous or time-averaged characteristics of neuronal activity, we focus on intermediate timescales-spike trains and waveforms of oscillatory potentials, which we consider as single entities. We find that, in healthy mice, spike arrangements and wave patterns (series of crests or troughs) are coupled to the animal's location, speed, and acceleration. In contrast, in tau-mice, neural activity is structurally disarrayed: brainwave cadence is detached from locomotion, spatial selectivity is lost, the spike flow is scrambled. Importantly, these alterations start early and accumulate with age, which exposes progressive disinvolvement the hippocampus circuit in spatial navigation. These features highlight qualitatively different neurodynamics than the ones provided by conventional analyses, and are more salient, thus revealing a new level of the hippocampal circuit disruptions.

Coherent olfactory bulb gamma oscillations arise from coupling independent columnar oscillators

Spike timing-based representations of sensory information depend on embedded dynamical frameworks within neuronal networks that establish the rules of local computation and interareal communication. Here, we investigated the dynamical properties of olfactory bulb circuitry in mice of both sexes using microelectrode array recordings from slice and in vivo preparations. Neurochemical activation or optogenetic stimulation of sensory afferents evoked persistent gamma oscillations in the local field potential. These oscillations arose from slower, GABA(A) receptor-independent intracolumnar oscillators coupled by GABA(A)-ergic synapses into a faster, broadly coherent network oscillation. Consistent with the theoretical properties of coupled-oscillator networks, the spatial extent of zero-phase coherence was bounded in slices by the reduced density of lateral interactions. The intact in vivo network, however, exhibited long-range lateral interactions that suffice in simulation to enable zero-phase gamma coherence across the olfactory bulb. The timing of action potentials in a subset of principal neurons was phase-constrained with respect to evoked gamma oscillations. Coupled-oscillator dynamics in olfactory bulb thereby enable a common clock, robust to biological heterogeneities, that is capable of supporting gamma-band spike synchronization and phase coding across the ensemble of activated principal neurons.NEW & NOTEWORTHY Odor stimulation evokes rhythmic gamma oscillations in the field potential of the olfactory bulb, but the dynamical mechanisms governing these oscillations have remained unclear. Establishing these mechanisms is important as they determine the biophysical capacities of the bulbar circuit to, for example, maintain zero-phase coherence across a spatially extended network, or coordinate the timing of action potentials in principal neurons. These properties in turn constrain and suggest hypotheses of sensory coding.

Detecting a periodic signal by a population of spiking neurons in the weakly nonlinear response regime

Motivated by experimental observations, we investigate a variant of the cocktail party problem: the detection of a weak periodic stimulus in the presence of fluctuations and another periodic stimulus which is stronger than the periodic signal to be detected. Specifically, we study the response of a population of stochastic leaky integrate-and-fire (LIF) neurons to two periodic signals and focus in particular on the question, whether the presence of one of the stimuli can be detected from the population activity. As a detection criterion, we use a simple threshold-crossing of the population activity over a certain time window. We show by means of the receiver operating characteristics (ROC) that the detectability depends only weakly on the time window of observation but rather strongly on the stimulus amplitude. Counterintuitively, the detection of the weak periodic signal can be facilitated by the presence of a strong periodic input current depending on the frequencies of the two signals and on the dynamical regime in which the neurons operate. Beside numerical simulations of the model, we present an analytical approximation for the ROC curve that is based on the weakly nonlinear response theory for a stochastic LIF neuron.

A biophysical and statistical modeling paradigm for connecting neural physiology and function

To understand single neuron computation, it is necessary to know how specific physiological parameters affect neural spiking patterns that emerge in response to specific stimuli. Here we present a computational pipeline combining biophysical and statistical models that provides a link between variation in functional ion channel expression and changes in single neuron stimulus encoding. More specifically, we create a mapping from biophysical model parameters to stimulus encoding statistical model parameters. Biophysical models provide mechanistic insight, whereas statistical models can identify associations between spiking patterns and the stimuli they encode. We used public biophysical models of two morphologically and functionally distinct projection neuron cell types: mitral cells (MCs) of the main olfactory bulb, and layer V cortical pyramidal cells (PCs). We first simulated sequences of action potentials according to certain stimuli while scaling individual ion channel conductances. We then fitted point process generalized linear models (PP-GLMs), and we constructed a mapping between the parameters in the two types of models. This framework lets us detect effects on stimulus encoding of changing an ion channel conductance. The computational pipeline combines models across scales and can be applied as a screen of channels, in any cell type of interest, to identify ways that channel properties influence single neuron computation.

Optimal phase-selective entrainment of heterogeneous oscillator ensembles

We develop a framework to design optimal entrainment signals that entrain an ensemble of heterogeneous nonlinear oscillators, described by phase models, at desired phases. We explicitly take into account heterogeneity in both oscillation frequency and the type of oscillators characterized by different Phase Response Curves. The central idea is to leverage the Fourier series representation of periodic functions to decode a phase-selective entrainment task into a quadratic program. We demonstrate our approach using a variety of phase models, where we entrain the oscillators into distinct phase patterns. Also, we show how the generalizability gained from our formulation enables us to meet a wide range of design objectives and constraints, such as minimum-power, fast entrainment, and charge-balanced controls.

Heterogeneity-induced synchronization in delay-coupled electronic oscillators

We study synchronization in networks of delay-coupled electronic oscillators, so-called phase-locked loops (PLLs). Using a phase-model description, we study the collective dynamics of mutually coupled PLLs and report the phenomenon of heterogeneity-induced synchronization. This phenomenon refers to the observation that heterogeneity in the system's parameters can induce synchronization by stabilizing the states which are unstable without such heterogeneity. In systems where component heterogeneity can be tuned and controlled, we show how the complex collective self-organized dynamics can be guided towards synchronized states with specific operational frequencies and phase relations. This is of importance for the technical applicability of self-organized dynamics. In electrical engineering, for example, where components can be strongly heterogeneous, our theoretical framework can inform the design process for networks of spatially distributed PLLs. The results presented here are also useful in understanding the collective dynamics in ensembles of phase oscillators with time-delayed interactions, inertia, and heterogeneity.

Cell and circuit origins of fast network oscillations in the mammalian main olfactory bulb

Neural synchrony generates fast network oscillations throughout the brain, including the main olfactory bulb (MOB), the first processing station of the olfactory system. Identifying the mechanisms synchronizing neurons in the MOB will be key to understanding how network oscillations support the coding of a high-dimensional sensory space. Here, using paired recordings and optogenetic activation of glomerular sensory inputs in MOB slices, we uncovered profound differences in principal mitral cell (MC) vs. tufted cell (TC) spike-time synchrony: TCs robustly synchronized across fast- and slow-gamma frequencies, while MC synchrony was weaker and concentrated in slow-gamma frequencies. Synchrony among both cell types was enhanced by shared glomerular input but was independent of intraglomerular lateral excitation. Cell-type differences in synchrony could also not be traced to any difference in the synchronization of synaptic inhibition. Instead, greater TC than MC synchrony paralleled the more periodic firing among resonant TCs than MCs and emerged in patterns consistent with densely synchronous network oscillations. Collectively, our results thus reveal a mechanism for parallel processing of sensory information in the MOB via differential TC vs. MC synchrony, and further contrast mechanisms driving fast network oscillations in the MOB from those driving the sparse synchronization of irregularly firing principal cells throughout cortex.

Molecular characterization of projection neuron subtypes in the mouse olfactory bulb

Projection neurons (PNs) in the mammalian olfactory bulb (OB) receive input from the nose and project to diverse cortical and subcortical areas. Morphological and physiological studies have highlighted functional heterogeneity, yet no molecular markers have been described that delineate PN subtypes. Here, we used viral injections into olfactory cortex and fluorescent nucleus sorting to enrich PNs for high-throughput single nucleus and bulk RNA deep sequencing. Transcriptome analysis and RNA in situ hybridization identified distinct mitral and tufted cell populations with characteristic transcription factor network topology, cell adhesion, and excitability-related gene expression. Finally, we describe a new computational approach for integrating bulk and snRNA-seq data and provide evidence that different mitral cell populations preferentially project to different target regions. Together, we have identified potential molecular and gene regulatory mechanisms underlying PN diversity and provide new molecular entry points into studying the diverse functional roles of mitral and tufted cell subtypes.

Traveling waves in non-local pulse-coupled networks

Traveling phase waves are commonly observed in recordings of the cerebral cortex and are believed to organize behavior across different areas of the brain. We use this as motivation to analyze a one-dimensional network of phase oscillators that are nonlocally coupled via the phase response curve (PRC) and the Dirac delta function. Existence of waves is proven and the dispersion relation is computed. Using the theory of distributions enables us to write and solve an associated stability problem. First and second order perturbation theory is applied to get analytic insight and we show that long waves are stable while short waves are unstable. We apply the results to PRCs that come from mitral neurons. We extend the results to smooth pulse-like coupling by reducing the nonlocal equation to a local one and solving the associated boundary value problem.

Synaptic Diversity Suppresses Complex Collective Behavior in Networks of Theta Neurons

Comprehending how the brain functions requires an understanding of the dynamics of neuronal assemblies. Previous work used a mean-field reduction method to determine the collective dynamics of a large heterogeneous network of uniformly and globally coupled theta neurons, which are a canonical formulation of Type I neurons. However, in modeling neuronal networks, it is unreasonable to assume that the coupling strength between every pair of neurons is identical. The goal in the present work is to analytically examine the collective macroscopic behavior of a network of theta neurons that is more realistic in that it includes heterogeneity in the coupling strength as well as in neuronal excitability. We consider the occurrence of dynamical structures that give rise to complicated dynamics via bifurcations of macroscopic collective quantities, concentrating on two biophysically relevant cases: (1) predominantly excitable neurons with mostly excitatory connections, and (2) predominantly spiking neurons with inhibitory connections. We find that increasing the synaptic diversity moves these dynamical structures to distant extremes of parameter space, leaving simple collective equilibrium states in the physiologically relevant region. We also study the node vs. focus nature of stable macroscopic equilibrium solutions and discuss our results in the context of recent literature.

Predicting responses to inhibitory synaptic input in substantia nigra pars reticulata neurons

The changes in firing probability produced by a synaptic input are usually visualized using the poststimulus time histogram (PSTH). It would be useful if postsynaptic firing patterns could be predicted from patterns of afferent synaptic activation, but attempts to predict the PSTH from synaptic potential waveforms using reasoning based on voltage trajectory and spike threshold have not been successful, especially for inhibitory inputs. We measured PSTHs for substantia nigra pars reticulata (SNr) neurons inhibited by optogenetic stimulation of striato-nigral inputs or by matching artificial inhibitory conductances applied by dynamic clamp. The PSTH was predicted by a model based on each SNr cell's phase-resetting curve (PRC). Optogenetic activation of striato-nigral input or artificial synaptic inhibition produced a PSTH consisting of an initial depression of firing followed by oscillatory increases and decreases repeating at the SNr cell's baseline firing rate. The phase resetting model produced PSTHs closely resembling the cell data, including the primary pause in firing and the oscillation. Key features of the PSTH, including the onset rate and duration of the initial inhibitory phase, and the subsequent increase in firing probability could be explained from the characteristic shape of the SNr cell's PRC. The rate of damping of the late oscillation was explained by the influence of asynchronous phase perturbations producing firing rate jitter and wander. Our results demonstrate the utility of phase-resetting models as a general method for predicting firing in spontaneously active neurons and their value in interpretation of the striato-nigral PSTH. NEW & NOTEWORTHY The coupling of patterned presynaptic input to sequences of postsynaptic firing is a Gordian knot, complicated by the multidimensionality of neuronal state and the diversity of potential initial states. Even so, it is fundamental for even the simplest understanding of network dynamics. We show that a simple phase-resetting model constructed from experimental measurements can explain and predict the sequence of spike rate changes following synaptic inhibition of an oscillating basal ganglia output neuron.

A coupled-oscillator model of olfactory bulb gamma oscillations

The olfactory bulb transforms not only the information content of the primary sensory representation, but also its underlying coding metric. High-variance, slow-timescale primary odor representations are transformed by bulbar circuitry into secondary representations based on principal neuron spike patterns that are tightly regulated in time. This emergent fast timescale for signaling is reflected in gamma-band local field potentials, presumably serving to efficiently integrate olfactory sensory information into the temporally regulated information networks of the central nervous system. To understand this transformation and its integration with interareal coordination mechanisms requires that we understand its fundamental dynamical principles. Using a biophysically explicit, multiscale model of olfactory bulb circuitry, we here demonstrate that an inhibition-coupled intrinsic oscillator framework, pyramidal resonance interneuron network gamma (PRING), best captures the diversity of physiological properties exhibited by the olfactory bulb. Most importantly, these properties include global zero-phase synchronization in the gamma band, the phase-restriction of informative spikes in principal neurons with respect to this common clock, and the robustness of this synchronous oscillatory regime to multiple challenging conditions observed in the biological system. These conditions include substantial heterogeneities in afferent activation levels and excitatory synaptic weights, high levels of uncorrelated background activity among principal neurons, and spike frequencies in both principal neurons and interneurons that are irregular in time and much lower than the gamma frequency. This coupled cellular oscillator architecture permits stable and replicable ensemble responses to diverse sensory stimuli under various external conditions as well as to changes in network parameters arising from learning-dependent synaptic plasticity.

Intraglomerular gap junctions enhance interglomerular synchrony in a sparsely connected olfactory bulb network

KEY POINTS

Despite sparse connectivity, population-level interactions between mitral cells (MCs) and granule cells (GCs) can generate synchronized oscillations in the rodent olfactory bulb. Intraglomerular gap junctions between MCs at the same glomerulus can greatly enhance synchronized activity of MCs at different glomeruli. The facilitating effect of intraglomerular gap junctions on interglomerular synchrony is through triggering of mutually synchronizing interactions between MCs and GCs. Divergent connections between MCs and GCs make minimal direct contribution to synchronous activity.

ABSTRACT

A dominant feature of the olfactory bulb response to odour is fast synchronized oscillations at beta (15-40 Hz) or gamma (40-90 Hz) frequencies, thought to be involved in integration of olfactory signals. Mechanistically, the bulb presents an interesting case study for understanding how beta/gamma oscillations arise. Fast oscillatory synchrony in the activity of output mitral cells (MCs) appears to result from interactions with GABAergic granule cells (GCs), yet the incidence of MC-GC connections is very low, around 4%. Here, we combined computational and experimental approaches to examine how oscillatory synchrony can nevertheless arise, focusing mainly on activity between 'non-sister' MCs affiliated with different glomeruli (interglomerular synchrony). In a sparsely connected model of MCs and GCs, we found first that interglomerular synchrony was generally quite low, but could be increased by a factor of 4 by physiological levels of gap junctional coupling between sister MCs at the same glomerulus. This effect was due to enhanced mutually synchronizing interactions between MC and GC populations. The potent role of gap junctions was confirmed in patch-clamp recordings in bulb slices from wild-type and connexin 36-knockout (KO) mice. KO reduced both beta and gamma local field potential oscillations as well as synchrony of inhibitory signals in pairs of non-sister MCs. These effects were independent of potential KO actions on network excitation. Divergent synaptic connections did not contribute directly to the vast majority of synchronized signals. Thus, in a sparsely connected network, gap junctions between a small subset of cells can, through population effects, greatly amplify oscillatory synchrony amongst unconnected cells.

Measurement of phase resetting curves using optogenetic barrage stimuli

BACKGROUND

The phase resetting curve (PRC) is a primary measure of a rhythmically firing neuron's responses to synaptic input, quantifying the change in phase of the firing oscillation as a function of the input phase. PRCs provide information about whether neurons will synchronize due to synaptic coupling or shared input. However, PRC estimation has been limited to in vitro preparations where stable intracellular recordings can be obtained and background activity is minimal, and new methods are required for in vivo applications.

NEW METHOD

We estimated PRCs using dense optogenetic stimuli and extracellular spike recording. Autonomously firing neurons in substantia nigra pars reticulata (SNr) of Thy1-channelrhodopsin 2 (ChR2) transgenic mice were stimulated with random barrages of light pulses, and PRCs were determined using multiple linear regression.

RESULTS

The PRCs obtained were type-I, showing only phase advances in response to depolarizing input, and generally sloped upward from early to late phases. Secondary PRCs, indicating the effect on the subsequent ISI, showed phase delays primarily for stimuli arriving at late phases. Phase models constructed from the optogenetic PRCs accounted for a large fraction of the variance in ISI length and provided a good approximation of the spike-triggered average stimulus.

COMPARISON WITH EXISTING METHODS

Compared to methods based on intracellular current injection, the new method sacrifices some temporal resolution. However, it should be much more widely applicable in vivo, because only extracellular recording and optogenetic stimulation are required.

CONCLUSIONS

These results demonstrate PRC estimation using methods suitable for in vivo applications.

Predicting the response of striatal spiny neurons to sinusoidal input

Spike-timing effects of small-amplitude sinusoidal currents were measured in mouse striatal spiny neurons firing repetitively. Spike-timing reliability varied with the stimulus frequency. For frequencies near the cell's firing rate, the cells altered firing rate to match the stimulus and became phase locked to it. The stimulus phase of firing during lock depended on the stimulus frequency relative to the cell's unperturbed firing rate. Interspike intervals during sinusoidal stimulation were predicted using an iterative map constructed from the cells' phase-resetting curve. Variability of interspike intervals was reduced by stimulation at all frequencies higher than about half the cell's unperturbed rate, and interspike intervals were accurately predicted by the map. Long sequences of spike times were predicted by iterating on the map. The accuracy of that prediction varied with frequency. Spike time predictability was highest near and during phase lock. The map predicted the phase of firing on the input and its dependence on stimulus frequency. Prediction errors, when they occurred, were of two kinds: unpredicted variation in interspike interval from intrinsic cell noise and accumulation of prediction errors from previous interspike intervals. Each type of prediction error arose from a different mechanism, and their impact was also predicted from the phase model. When two oscillatory input currents were presented simultaneously, striatal neurons responded selectively to only one of them, the one closest in frequency to the cell's unperturbed firing rate. Their spike times encoded the frequency and phase of that single oscillatory input.NEW & NOTEWORTHY During repetitive firing, the timing of action potentials is determined by the interaction between the input and voltage-sensitive currents throughout the interspike interval. This interaction is encapsulated in the neuron's phase-resetting curve. The phase-resetting curve predicted spike timing to small sinusoidal currents over a wide range of stimulus frequencies. Firing patterns were most sensitive to oscillatory components near the cell's own firing rate, even in the presence of noise and other inputs.

Synchronization of oscillators through time-shifted common inputs

Shared upstream dynamical processes are frequently the source of common inputs in various physical and biological systems. However, due to finite signal transmission speeds and differences in the distance to the source, time shifts between otherwise common inputs are unavoidable. Since common inputs can be a source of correlation between the elements of multi-unit dynamical systems, regardless of whether these elements are directly connected with one another or not, it is of importance to understand their impact on synchronization. As a canonical model that is representative for a variety of different dynamical systems, we study limit-cycle oscillators that are driven by stochastic time-shifted common inputs. We show that if the oscillators are coupled, time shifts in stochastic common inputs do not simply shift the distribution of the phase differences, but rather the distribution actually changes as a result. The best synchronization is therefore achieved at a precise intermediate value of the time shift, which is due to a resonance-like effect with the most probable phase difference that is determined by the deterministic dynamics.

Oscillating systems with cointegrated phase processes

We present cointegration analysis as a method to infer the network structure of a linearly phase coupled oscillating system. By defining a class of oscillating systems with interacting phases, we derive a data generating process where we can specify the coupling structure of a network that resembles biological processes. In particular we study a network of Winfree oscillators, for which we present a statistical analysis of various simulated networks, where we conclude on the coupling structure: the direction of feedback in the phase processes and proportional coupling strength between individual components of the system. We show that we can correctly classify the network structure for such a system by cointegration analysis, for various types of coupling, including uni-/bi-directional and all-to-all coupling. Finally, we analyze a set of EEG recordings and discuss the current applicability of cointegration analysis in the field of neuroscience.

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

Neurons as oscillators

Regularly spiking neurons can be described as oscillators. In this article we review some of the insights gained from this conceptualization and their relevance for systems neuroscience. First, we explain how a regularly spiking neuron can be viewed as an oscillator and how the phase-response curve (PRC) describes the response of the neuron's spike times to small perturbations. We then discuss the meaning of the PRC for a single neuron's spiking behavior and review the PRCs measured from a variety of neurons in a range of spiking regimes. Next, we show how the PRC can be related to a number of common measures used to quantify neuronal firing, such as the spike-triggered average and the peristimulus histogram. We further show that the response of a neuron to correlated inputs depends on the shape of the PRC. We then explain how the PRC of single neurons can be used to predict neural network behavior. Given the PRC, conduction delays, and the waveform and time course of the synaptic potentials, it is possible to predict neural population behavior such as synchronization. The PRC also allows us to quantify the robustness of the synchronization to heterogeneity and noise. We finally ask how to combine the measured PRCs and the predictions based on PRC to further the understanding of systems neuroscience. As an example, we discuss how the change of the PRC by the neuromodulator acetylcholine could lead to a destabilization of cortical network dynamics. Although all of these studies are grounded in mathematical abstractions that do not strictly hold in biology, they provide good estimates for the emergence of the brain's network activity from the properties of individual neurons. The study of neurons as oscillators can provide testable hypotheses and mechanistic explanations for systems neuroscience.

Diversity improves performance in excitable networks

As few real systems comprise indistinguishable units, diversity is a hallmark of nature. Diversity among interacting units shapes properties of collective behavior such as synchronization and information transmission. However, the benefits of diversity on information processing at the edge of a phase transition, ordinarily assumed to emerge from identical elements, remain largely unexplored. Analyzing a general model of excitable systems with heterogeneous excitability, we find that diversity can greatly enhance optimal performance (by two orders of magnitude) when distinguishing incoming inputs. Heterogeneous systems possess a subset of specialized elements whose capability greatly exceeds that of the nonspecialized elements. We also find that diversity can yield multiple percolation, with performance optimized at tricriticality. Our results are robust in specific and more realistic neuronal systems comprising a combination of excitatory and inhibitory units, and indicate that diversity-induced amplification can be harnessed by neuronal systems for evaluating stimulus intensities.

Stochastic Synchronization in Purkinje Cells with Feedforward Inhibition Could Be Studied with Equivalent Phase-Response Curves

Simple-spike synchrony between Purkinje cells projecting to a common neuron in the deep cerebellar nucleus is emerging as an important factor in the encoding of output information from cerebellar cortex. A phenomenon known as stochastic synchronization happens when uncoupled oscillators synchronize due to correlated inputs. Stochastic synchronization is a viable mechanism through which simple-spike synchrony could be generated, but it has received scarce attention, perhaps because the presence of feedforward inhibition in the input to Purkinje cells makes insights difficult. This paper presents a method to account for feedforward inhibition so the usual mathematical approaches to stochastic synchronization can be applied. The method consists in finding a single Phase Response Curve, called the equivalent PRC, that accounts for the effects of both excitatory inputs and delayed feedforward inhibition from molecular layer interneurons. The results suggest that a theory of stochastic synchronization for the case of feedforward inhibition may not be necessary, since this case can be approximately reduced to the case of inputs characterized by a single PRC. Moreover, feedforward inhibition could in many situations increase the level of synchrony experienced by Purkinje cells.

Postnatal development attunes olfactory bulb mitral cells to high-frequency signaling

Mitral cells (MCs) are a major class of principal neurons in the vertebrate olfactory bulb, conveying odor-evoked activity from the peripheral sensory neurons to olfactory cortex. Previous work has described the development of MC morphology and connectivity during the first few weeks of postnatal development. However, little is known about the postnatal development of MC intrinsic biophysical properties. To understand stimulus encoding in the developing olfactory bulb, we have therefore examined the development of MC intrinsic biophysical properties in acute slices from postnatal day (P)7-P35 mice. Across development, we observed systematic changes in passive membrane properties and action potential waveforms consistent with a developmental increase in sodium and potassium conductances. We further observed developmental decreases in hyperpolarization-evoked membrane potential sag and firing regularity, extending recent links between MC sag heterogeneity and firing patterns. We then applied a novel combination of statistical analyses to examine how the evolution of these intrinsic biophysical properties specifically influenced the representation of fluctuating stimuli by MCs. We found that immature MCs responded to frozen fluctuating stimuli with lower firing rates, lower spike-time reliability, and lower between-cell spike-time correlations than more mature MCs. Analysis of spike-triggered averages revealed that these changes in spike timing were driven by a developmental shift from broad integration of inputs to more selective detection of coincident inputs. Consistent with this shift, generalized linear model fits to MC firing responses demonstrated an enhanced encoding of high-frequency stimulus features by mature MCs.

Stabilizing synchrony by inhomogeneity

We show that for two weakly coupled identical neuronal oscillators with strictly positive phase resetting curve, isochronous synchrony can only be seen in the absence of noise and an arbitrarily weak noise can destroy entrainment and generate intermittent phase slips. Small inhomogeneity–mismatch in the intrinsic firing rate of the neurons–can stabilize the phase locking and lead to more precise relative spike timing of the two neurons. The results can explain how for a class of neuronal models, including leaky integrate-fire model, inhomogeneity can increase correlation of spike trains when the neurons are synaptically connected.

Olfactory system oscillations across phyla

Neural oscillations are ubiquitous in olfactory systems of mammals, insects and molluscs. Neurophysiological and computational investigations point to common mechanisms for gamma or odor associated oscillations across phyla (40-100Hz in mammals, 20-30Hz in insects, 0.5-1.5Hz in molluscs), engaging the reciprocal dendrodendritic synapse between excitatory principle neurons and inhibitory interneurons in the olfactory bulb (OB), antennal lobe (AL), or procerebrum (PrC). Recent studies suggest important mechanisms that may modulate gamma oscillations, including neuromodulators and centrifugal input to the OB and AL. Beta (20Hz) and theta (2-12Hz) oscillations coordinate activity within and across brain regions. Olfactory beta oscillations are associated with odor learning and depend on centrifugal OB input, while theta oscillations are strongly associated with respiration.

Olfactory dysfunction predicts 5-year mortality in older adults

Prediction of mortality has focused on disease and frailty, although antecedent biomarkers may herald broad physiological decline. Olfaction, an ancestral chemical system, is a strong candidate biomarker because it is linked to diverse physiological processes. We sought to determine if olfactory dysfunction is a harbinger of 5-year mortality in the National Social Life, Health and Aging Project [NSHAP], a nationally representative sample of older U.S. adults. 3,005 community-dwelling adults aged 57–85 were studied in 2005–6 (Wave 1) and their mortality determined in 2010–11 (Wave 2). Olfactory dysfunction, determined objectively at Wave 1, was used to estimate the odds of 5-year, all cause mortality via logistic regression, controlling for demographics and health factors. Mortality for anosmic older adults was four times that of normosmic individuals while hyposmic individuals had intermediate mortality (p<0.001), a “dose-dependent” effect present across the age range. In a comprehensive model that included potential confounding factors, anosmic older adults had over three times the odds of death compared to normosmic individuals (OR, 3.37 [95%CI 2.04, 5.57]), higher than and independent of known leading causes of death, and did not result from the following mechanisms: nutrition, cognitive function, mental health, smoking and alcohol abuse or frailty. Olfactory function is thus one of the strongest predictors of 5-year mortality and may serve as a bellwether for slowed cellular regeneration or as a marker of cumulative toxic environmental exposures. This finding provides clues for pinpointing an underlying mechanism related to a fundamental component of the aging process.

The Competing Benefits of Noise and Heterogeneity in Neural Coding

Noise and heterogeneity are both known to benefit neural coding. Stochastic resonance describes how noise, in the form of random fluctuations in a neuron's membrane voltage, can improve neural representations of an input signal. Neuronal heterogeneity refers to variation in any one of a number of neuron parameters and is also known to increase the information content of a population. We explore the interaction between noise and heterogeneity and find that their benefits to neural coding are not independent. Specifically, a neuronal population better represents an input signal when either noise or heterogeneity is added, but adding both does not always improve representation further. To explain this phenomenon, we propose that noise and heterogeneity operate using two shared mechanisms: (1) temporally desynchronizing the firing of neurons in the population and (2) linearizing the response of a population to a stimulus. We first characterize the effects of noise and heterogeneity on the information content of populations of either leaky integrate-and-fire or FitzHugh-Nagumo neurons. We then examine how the mechanisms of desynchronization and linearization produce these effects, and find that they work to distribute information equally across all neurons in the population in terms of both signal timing (desynchronization) and signal amplitude (linearization). Without noise or heterogeneity, all neurons encode the same aspects of the input signal; adding noise or heterogeneity allows neurons to encode complementary aspects of the input signal, thereby increasing information content. The simulations detailed in this letter highlight the importance of heterogeneity and noise in population coding, demonstrate their complex interactions in terms of the information content of neurons, and explain these effects in terms of underlying mechanisms.

Disrupting information coding via block of 4-AP-sensitive potassium channels

Recent interest has emerged on the role of intrinsic biophysical diversity in neuronal coding. An important question in neurophysiology is understanding which voltage-gated ion channels are responsible for this diversity and how variable expression or activity of one class of ion channels across neurons of a single type affects they way populations carry information. In mitral cells in the olfactory bulb of mice, we found that biophysical diversity was conferred in part by 4-aminopyridine (4-AP)-sensitive potassium channels and reduced following block of those channels. When populations of mitral cells were stimulated with identical inputs, the diversity exhibited in their output spike patterns reduced with the addition of 4-AP, decreasing the stimulus information carried by ensembles of 15 neurons from 437 ± 15 to 397 ± 19 bits/s. Decreases in information were due to reduction in the diversity of population spike patterns generated in response to different features of the stimulus, suggesting that the coding capacity of a population can be altered by changes in the function of single ion channel types.

Greater excitability and firing irregularity of tufted cells underlies distinct afferent-evoked activity of olfactory bulb mitral and tufted cells

Mitral and tufted cells, the two classes of principal neurons in the mammalian main olfactory bulb, exhibit morphological differences but remain widely viewed as functionally equivalent. Results from several recent studies, however, suggest that these two cell classes may encode complementary olfactory information in their distinct patterns of afferent-evoked activity. To understand how these differences in activity arise, we have performed the first systematic comparison of synaptic and intrinsic properties between mitral and tufted cells. Consistent with previous studies, we found that tufted cells fire with higher probability and rates and shorter latencies than mitral cells in response to physiological afferent stimulation. This stronger response of tufted cells could be partially attributed to synaptic differences, as tufted cells received stronger afferent-evoked excitation than mitral cells. However, differences in intrinsic excitability also contributed to the differences between mitral and tufted cell activity. Compared to mitral cells, tufted cells exhibited twofold greater excitability and peak instantaneous firing rates. These differences in excitability probably arise from differential expression of voltage-gated potassium currents, as tufted cells exhibited faster action potential repolarization and afterhyperpolarizations than mitral cells. Surprisingly, mitral and tufted cells also showed firing mode differences. While both cell classes exhibited regular firing and irregular stuttering of action potential clusters, tufted cells demonstrated a greater propensity to stutter than mitral cells. Collectively, stronger afferent-evoked excitation, greater intrinsic excitability and more irregular firing in tufted cells can combine to drive distinct responses of mitral and tufted cells to afferent-evoked input.

Spectral reconstruction of phase response curves reveals the synchronization properties of mouse globus pallidus neurons

The propensity of a neuron to synchronize is captured by its infinitesimal phase response curve (iPRC). Determining whether an iPRC is biphasic, meaning that small depolarizing perturbations can actually delay the next spike, if delivered at appropriate phases, is a daunting experimental task because negative lobes in the iPRC (unlike positive ones) tend to be small and may be occluded by the normal discharge variability of a neuron. To circumvent this problem, iPRCs are commonly derived from numerical models of neurons. Here, we propose a novel and natural method to estimate the iPRC by direct estimation of its spectral modes. First, we show analytically that the spectral modes of the iPRC of an arbitrary oscillator are readily measured by applying weak harmonic perturbations. Next, applying this methodology to biophysical neuronal models, we show that a low-dimensional spectral reconstruction is sufficient to capture the structure of the iPRC. This structure was preserved reasonably well even with added physiological scale jitter in the neuronal models. To validate the methodology empirically, we applied it first to a low-noise electronic oscillator with a known design and then to cortical pyramidal neurons, recorded in whole cell configuration, that are known to possess a monophasic iPRC. Finally, using the methodology in conjunction with perforated-patch recordings from pallidal neurons, we show, in contrast to recent modeling studies, that these neurons have biphasic somatic iPRCs. Biphasic iPRCs would cause lateral somatically targeted pallidal inhibition to desynchronize pallidal neurons, providing a plausible explanation for their lack of synchrony in vivo.

Impact of neuronal heterogeneity on correlated colored noise-induced synchronization

Synchronization plays an important role in neural signal processing and transmission. Many hypotheses have been proposed to explain the origin of neural synchronization. In recent years, correlated noise-induced synchronization has received support from many theoretical and experimental studies. However, many of these prior studies have assumed that neurons have identical biophysical properties and that their inputs are well modeled by white noise. In this context, we use colored noise to induce synchronization between oscillators with heterogeneity in both phase-response curves and frequencies. In the low noise limit, we derive novel analytical theory showing that the time constant of colored noise influences correlated noise-induced synchronization and that oscillator heterogeneity can limit synchronization. Surprisingly, however, heterogeneous oscillators may synchronize better than homogeneous oscillators given low input correlations. We also find resonance of oscillator synchronization to colored noise inputs when firing frequencies diverge. Collectively, these results prove robust for both relatively high noise regimes and when applied to biophysically realistic spiking neuron models, and further match experimental recordings from acute brain slices.

Active decorrelation in the basal ganglia

The cytoarchitecturally-homogeneous appearance of the globus pallidus, subthalamic nucleus and substantia nigra has long been said to imply a high degree of afferent convergence and sharing of inputs by nearby neurons. Moreover, axon collaterals of neurons in the external segment of the globus pallidus and the substantia nigra pars reticulata arborize locally and make inhibitory synapses on other cells of the same type. These features suggest that the connectivity of the basal ganglia may impose spike-time correlations among the cells, and it has been puzzling that experimental studies have failed to demonstrate such correlations. One possible solution arises from studies of firing patterns in basal ganglia cells, which reveal that they are nearly all pacemaker cells. Their high rate of firing does not depend on synaptic excitation, but they fire irregularly because a dense barrage of synaptic inputs normally perturbs the timing of their autonomous activity. Theoretical and computational studies show that the responses of repetitively-firing neurons to shared input or mutual synaptic coupling often defy classical intuitions about temporal synaptic integration. The patterns of spike-timing among such neurons depend on the ionic mechanism of pacemaking, the level of background uncorrelated cellular and synaptic noise, and the firing rates of the neurons, as well as the properties of their synaptic connections. Application of these concepts to the basal ganglia circuitry suggests that the connectivity and physiology of these nuclei may be configured to prevent the establishment of permanent spike-timing relationships between neurons. The development of highly synchronous oscillatory patterns of activity in Parkinson's disease may result from the loss of pacemaking by some basal ganglia neurons, and accompanying breakdown of the mechanisms responsible for active decorrelation.

Stimulus features, resetting curves, and the dependence on adaptation

We derive a formula that relates the spike-triggered covariance (STC) to the phase resetting curve (PRC) of a neural oscillator. We use this to show how changes in the shape of the PRC alter the sensitivity of the neuron to different stimulus features, which are the eigenvectors of the STC. We compute the PRC and STC for some biophysical models. We compare the STCs and their spectral properties for a two-parameter family of PRCs. Surprisingly, the skew of the PRC has a larger effect on the spectrum and shape of the STC than does the bimodality of the PRC (which plays a large role in synchronization properties). Finally, we relate the STC directly to the spike-triggered average and apply this theory to an olfactory bulb mitral cell recording.